description/proof of that metric is continuous
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of metric.
- The reader admits the proposition that any metric is continuous with respect to the topology induced by the metric.
- The reader admits the proposition that for any finite-product metric space, the topology induced by the product metric is the product topology of the topologies induced by the constituent metrics.
- The reader admits the proposition that for any metric spaces map, the map is continuous at any point if and only if the map is continuous at the point between the induced topological spaces.
Target Context
- The reader will have a description and a proof of the proposition that any metric is continuous.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(M\): \(\in \{\text{ the metric spaces }\}\), with any metric, \(dist\)
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Statements:
\(dist \in \{\text{ the continuous, metric spaces maps }\}\)
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2: Note
Of course, this proposition should be able to be proved directly as a metric spaces map, but as we already have the continuousness as the induced topological spaces map, we use it.
3: Proof
Whole Strategy: Step 1: see that \(dist\) is continuous as the induced topological spaces map; Step 2: conclude the proposition.
Step 1:
\(dist: M \times M \to \mathbb{R}\) is continuous with \(M \times M\) regarded as the topological space with the topology as the product of the topology for \(M\) induced by \(dist\), by the proposition that any metric is continuous with respect to the topology induced by the metric.
Step 2:
The product topology is the topology induced by the product metric, by the proposition that for any finite-product metric space, the topology induced by the product metric is the product topology of the topologies induced by the constituent metrics.
The Euclidean topological space, \(\mathbb{R}\), has the topology induced by the Euclidean metric, by definition.
So, \(dist\) as the metric spaces map is continuous, by the proposition that for any metric spaces map, the map is continuous at any point if and only if the map is continuous at the point between the induced topological spaces.
